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arXiv:1002.5023

The nonlinear thermodynamic quantum master equation

Hans Christian Öttinger

Published Feb 26, 2010• 5 citations
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Abstract

The quantum master equation obtained by generalizing the geometric formulation of nonequilibrium thermodynamics to dissipative quantum systems is seriously nonlinear. We argue that nonlinearity occurs naturally in the step from reversible to irreversible equations and we analyze the nature of the nonlinear contribution. The thermodynamic nonlinearity leads to proper equilibrium solutions and improves the dynamic behavior of dissipative quantum systems. We also discuss the Markovian character of the thermodynamic quantum master equation. The general ideas are illustrated for two-level systems.

DOI
10.48550/arxiv.1002.5023

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The quantum master equation obtained by generalizing the geometric formulation of nonequilibrium thermodynamics to dissipative quantum systems is seriously nonlinear. We argue that nonlinearity occurs naturally in the step from reversible to irreversible equations and we analyze the nature of the nonlinear contribution. The thermodynamic nonlinearity leads to proper equilibrium solutions and improves the dynamic behavior of dissipative quantum systems. We also discuss the Markovian character of the thermodynamic quantum master equation. The general ideas are illustrated for two-level systems.
The quantum master equation obtained from two different thermodynamic arguments is seriously nonlinear. We argue that, for quantum systems, nonlinearity occurs naturally in the step from reversible to irreversible equations and we analyze the nature and consequences of the nonlinear contribution. The thermodynamic nonlinearity naturally leads to canonical equilibrium solutions and extends the range of validity to lower temperatures. We discuss the Markovian character of the thermodynamic quantum master equation and introduce a solution strategy based on coupled evolution equations for the eigenstates and eigenvalues of the density matrix. The general ideas are illustrated for the two-level system and for the damped harmonic oscillator. Several conceptual implications of the nonlinearity of the thermodynamic quantum master equation are pointed out, including the absence of a Heisenberg picture and the resulting difficulties with defining multitime correlations.
In this paper, we outline the main features of the nonlinear quantum evolution equation proposed by the present author. Such an equation may be used as a model of reduced subsystem dynamics to complement various historical and contemporary efforts to extend linear Markovian theories of dissipative phenomena and relaxation based on master equations, Lindblad and Langevin equations, to the nonlinear and far nonequilibrium domain. It may also be used as the fundamental dynamical principle in theories that attempt to unite mechanics and thermodynamics, such as the Hatsopoulos–Gyftopoulos unified theory which motivated the original development of this well-behaved general nonlinear equation for the evolution of the density operator capable of generating irreversible deterministic relaxation to thermodynamic equilibrium from any far nonequilibrium state even for an isolated system.
We investigate the zero-temperature limit of thermodynamic quantum master equations that govern the time evolution of density matrices for dissipative quantum systems. The quantum master equations for T=0T=0T=0 and for T>0T>0T>0 possess completely different structures: (i) the equation for T=0T=0T=0 is linear in the deviation from the ground-state density matrix, whereas the equation for T>0T>0T>0, in general, is seriously nonlinear, and (ii) the Gibbs state is obtained as the steady-state solution of the nonlinear equation for T>0T>0T>0, whereas the ground state cannot be found from the equation for T=0T=0T=0. Nevertheless, the equation for T=0T=0T=0 can reproduce the behavior for $T\ensuremath{\gtrsim}0$ remarkably well. We discuss some implications of that observation for dissipative quantum field theory.
Dirac's method of classical analogy is employed to include quantum degrees of freedom into a geometric framework of nonequilibrium thermodynamics. The proposed formulation of dissipative quantum mechanics builds entirely upon the geometric structures implied by commutators and canonical correlations. A lucid formulation of a nonlinear quantum master equation follows from the thermodynamic structure. The approach is applicable even at very low temperatures and complex classical environments with internal structure can be handled readily.
We simulate a piecewise deterministic Markovian jump process that represents an unraveling of a nonlinear quantum master equation describing the evolution of a quantum subsystem in contact with a heat bath. This process relies on running ensemble averages and state vectors normalized only on average. The latter causes the normalization of a small subset of trajectories to grow exponentially, resulting in numerical instability. We propose an efficient solution to this problem and illustrate the general ideas for a harmonic oscillator and a two-level system, each of them being weakly coupled to a heat reservoir. Our findings lay the foundation for a powerful simulation technique for dissipative quantum systems surrounded by general classical nonequilibrium environments.
We develop a general theory describing the thermodynamical behavior of open quantum systems coupled to thermal baths beyond perturbation theory. Our approach is based on the exact time-local quantum master equation for the reduced open-system states, and on a principle of minimal dissipation. This principle leads to a unique prescription for the decomposition of the master equation into a Hamiltonian part representing coherent time evolution and a dissipator part describing dissipation and decoherence. Employing this decomposition we demonstrate how to define work, heat, and entropy production, formulate the first and second laws of thermodynamics, and establish the connection between violations of the second law and quantum non-Markovianity.